I learned this interesting theorem but cannot remember from where since a long time has passed. What I'm wondering now is: Does someone know some constants which this theorem holds? ie, give some values of $a$, such that $[a^n]$ is always prime for $n\in N^+$(where $[\cdots]$ denotes integral part of a real number and $N^+$ is the set of positive integers).

$\begingroup$As Sylvin Julien points out in an answer, the prime-producing formula is doubly exponential: $[a^{3^n}]$ (or $[a^{c^n}]$) for all $n$. It is a curiosity but I would not call it an interesting theorem because the proof shows it's a rigged formula: you need to use primes to get the formula and it's totally unrealistic to think there is some independent way to describe the number $a$ without using primes. It's uselessness is similar in spirit to the "formula" for counting primes in math.stackexchange.com/questions/776997/….$\endgroup$
– KConradAug 24 '17 at 14:59

2 Answers
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According to Dubickas, the problem was still open in 2009; the conjecture is that no such $\alpha$ exists. Dubickas and his collaborators have worked extensively on integer parts of powers. See also Baker&Harman for a similar problem and more results in the negative direction.

$\begingroup$I should add that this is comparatively much harder than Mills's theorem, just as Mills's relies on much deeper results than for Wright's constant (spacing between successive primes vs. Bertrand's postulate); an argument for not expecting such an $\alpha$ is that if you run an analogous argument you need an estimate on primes which is too good to be true.$\endgroup$
– Emanuele TronAug 24 '17 at 15:52

Maybe not exactly what you're looking for, but in his book "Merveilleux nombres premiers" (in French), Jean-Paul Delahaye mentions the so-called Mills' constant $A=1.30637788386...$ which fulfills, under the assumption of the Riemann Hypothesis, the property that for all positive integer $n$, the integral part of $A^{3^{n}}$ is a prime.

$\begingroup$Worth mentioning that such a constant $A$ is known to exist unconditionally. Whether there is one which is approximately $1.306\dots$ is conditional on RH.$\endgroup$
– WojowuAug 24 '17 at 16:09